DTris,10−90%/Tw

0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.1 0.2 0.3 0.4 0.5 0.6

PSfragreplaements

Overshoot

D

O S [% ]

Figure 5.22: Behavior of eah step response harateristi parameter with

re-spetto the damping ratio

Sinus response test

Aording tothistest,thebehaviorofanatuatorinthefrequenydomainan

beidentied. Theinputforsuhatestisasinusoidalsignalwithaninreasing

frequeny and adened amplitude.

TheBode diagramisthe mostimportantharateristiofthis test,gure5.23.

Thedamping ratioandthetime onstant arethesigniantparameters

hang-ing the bode diagram of the seond-order transfer funtion. The eet of the

damping ratio anbesummarizedasfollows: ifthedamping ratioof a

seond-order transfer funtion is less than 0.707, a resonane ours at the natural

frequeny of the system. The natural frequeny is

ω ₀ = _T ¹

w

. Inversely, if the

damping ratio islargerthan0.707, no resonaneours. Theeet ofthetime

onstant (

T w

^{) is on the plae where the bode diagram falls o. By inreasing}

T _w

^{,the falling happensinthe low} frequenies.

Extension of the methodology

Aording to theV-model, gure5.1, therequirementsof a large-sale system

an be broken down into the requirements formulated on the design variables

ofomponents(System Design). Withtheintrodution oftheveriation tests

in the last setion, the dependeny graph will be extended. In the extended

dependeny graph, a new level is added, the so-alled omponent design.

A-ordingly,the so-alledVeriationVariables level, whereveriationvariables

of eah veriation test are indiated, are dened in the same level of the

Subsystem level in the original V-model, gure 5.24 . Below this, the physial

Components Details areonsidered, whihshouldbespeied bythe

manufa-turer. In this way, we formulate the requirements on theveriation variables

from the requirements of the large-sale system. The way inwhih the

atua-tor isonstruted physiallywillbe surrendered tothe manufaturer. But the

0.3 0.5 1 2 3 4 -6

-5 -4 -3 -2 -1 0

0.3 0.5 1 2 3 4

-100 -50 0

PSfragreplaements

Frequeny [Hz℄

Frequeny [Hz℄

Magnitude[dB℄

Phasein[

◦

℄

Figure 5.23: Bode diagram of a transfer funtion witha damping ratio larger

than 0.707

manufaturer annotdeliveranykindofphysialatuators, rather therealized

atuator shouldbeinthe form ofthe predeessors.

Figure5.24: Extended V-Modelfor thedesign ofatuators

Summary

Inthis hapter, we presentedthe robustdesign methodfordesigningatuators

and ontrol systems, whih an ontrol the omplexities in the development

proess and nd optimized intervals for the design variables,rather than only

one optimal set of the design variables. We have shown that the method is

based on the so-alled V-model in the system engineering. The method also

inludesthreeenablers,whiharedependeny graph,Bottom-Upmappingand

Top-Down mapping. Thedependenygraph manages theknow-howinthe

de-sign proedure andisagraphinwhihthe measurable system,subsystem,and

omponent properties are depited as verties. Their dependeniesor

intera-tions were shown withan edge, whose diretion shows whih subsystems' and

omponents' properties interat with eah other and inuene the measurable

system properties. The Bottom-Up mapping is thequantitative evaluation of

theinueneoftheinputsontheoutputs. Forthispurpose,weneededamodel

whih maps the inputs to the outputs. This model wasthe atuator and the

ontrol system model developed in the last two hapters. We explained the

neessity of lots of simulations on this model, whih provide information for

the Top-Down mapping. As a onsequene, we have laried the idea of the

Design of Experiments (DoEs) and how we an nd the best method ofDoEs

based on the disrepany fator. We also used the artiial neural networks

and support vetor mahines as meta-models to model our original atuator

and ontrol systemmodel. Thiswasdonetoaelerate theoptimization

proe-dure. Thisoptimizationproblemwasto ndthelargestbox, so-alledsolution

box, inside of the solution spae of the design variables. The solution spae

integrated dierent requirements fromdiverse disiplines. The edgesof the

so-lution boxservedasindependenttarget regionsforomponent properties. The

solution boxalso enhaned the probability of nding a valid design in the

so-lution spae. Welariedthatwehaveto onverttherequirementsformulated

on the design variables into the requirements whih should be formulated on

theveriation variables. Asaonsequene,weexplained dierent veriation

tests andvariables for thedesign variables ofan atuator.

6 Appliation

6.1 Results for robust designing of the rear steering

system

Inthis hapter, we applytheexplainedmethodintheprevioushapterfor

for-mulating requirementsonthedesignvariablesoftheontrolsystem,introdued

inhapter 4,andthe design variablesof ARSatuators, introdued inhapter

5. The rst step is to determine the dependeny graph. In this dependeny

graph, the relations between design variables of ARS atuator and theontrol

system and the objetive driving targets (CVs) are demonstrated. Moreover,

therelationsbetweenCVsandtheassessmentindies(AIs)aredepited,gure

6.1 .

Figure 6.1:Dependenygraphforthe ARSatuator,thelateraldynamis

on-trol system, andthe vehile genes

Asanbeseen,thedesignvariablesofeahfuntionoftheontrolsystemand

the ARS atuator and the genes of the vehile inuene the determined CVs.

For example, all design variables of thestati feedforward ontrol of ARS an

only eet the stationary objetive driving dynamis performane measures,

namely BRWCand QSSC.

Identiation of design variables

In the seond step, we have to identify the variables of the ontrol system

and the ARS atuator on whih the requirements have to be formulated. The

relevant parameters areasfollows:

Table6.1: Designvariablesof theARS atuator andits assoiatedontrol

sys-tem

System Parameter Desription

v _trans v _max i ARS,stat.,min

Stat. Feedforward

i ARS,stat.,max

Charateristi urve forthe stationary

feedforward (gure 3.2)

T _f ω _f

Dyn. Feedforward

D _f

Fators forthedesired transferfuntion

G ψδ ˙ f ,des

^(equation??)

Feedbak

K _p

^{Feedbakontroller} proportionalgain (equation 3.45 )

K p

D T _w T _d

Atuator LTItransferfuntion paramters

(equation 4.1 )

F _max v _noload v _{(F max )}

Atuator fore-speed harateristi urve

(gure 6.2 ) Atuator

δ _r,max

^{Maximum permitted rearsteering wheelangle}

m

^Vehilemass

rearload

^{Vehilerear loadratio}

h

^{VehileCG-height}

Genes

J _z

^{VehileInertia inz axis}

It should be onsidered that the harateristi urve of theatuator has to

have aunitformlikethe one ingure6.2forthedesign regardingthe

ounter-Aordingly, we deal with three dierent groups of design variables. The

rst groupontains the genes of a vehile introdued in table 6.1 . Theyhave

their nominalvalueswhiharealreadypredened. A marginisleftfor themin

order toensuretherobustnessofthedesignedARSatuator anditsassoiated

ontrol system logi with respet to the hanges inthe vehile genes whih is

the ase, if we deal with dierent derivatives. The seond group inludes the

design variables oftheARS atuatorandthe thirdgroup ismadeofthedesign

variables ofthe ontrol systemlogi relevant totheARS.

Figure6.2: QSSC-CV

Forallthesedesignvariables,werstsamplethedesignspae. Thesampling

method is sramble Halton sequene in this appliation and the number of

sampling pointsis 4000.

Theonsidereddriving dynamis performane measures(CVs)arelisted

be-low:

Table6.2: Considered objetivedriving dynamis performane measures

Maneuver CV

EG _H,nl

QSSC

a _y,max

( ˙ ψ/δ _H ) _stat,70 ( ˙ ψ/δ _H ) _stat,190

WEAVE

( ˙ ψ/δ _H ) _stat,max β _max

F _z,min

SWD

SF _{M AX} T _{eq, ψ/δ ˙}

H

(H/H ₀ ) _{ψ/δ ˙}

H

CSST

T _{a y /δ H}

BRWC

∆ ˙ ψ _1s

For eah ofthese CVs, the simulation of the whole systemis exeuted 4000

times. Then, individual ANNand SVM are trained byeah output (CV)and

a set of inputs (designvariables). Some results of ANN training areshown in

gure 6.3and gure6.4.

0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34

0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34

PSfrag replaements

WEAVE -

( ψ/δ ˙ _H ) _stat,max

Target

Neuralnetworkpredition

(a)

R ² _test

^=0.985

R _training ²

^=0.990

-4 -3 -2 -1 0 1 2 3 4

-4 -3 -2 -1 0 1 2 3 4

PSfrag replaements

BRWC

-ψ ˙ _∆1s

Target

Neuralnetworkpredition

(b)

R ² _test

^=0.920

R ² _training

^=0.949

Figure6.3: Regression plots for CV

ψ ˙ _∆1s

^and

( ˙ ψ/δ _H ) _stat,max

8 8.5 9 9.5 10 8

8.5 9 9.5 10

PSfrag replaements

QSSC -

a _y,max

Target

Neuralnetworkpredition

(a)

R ² _test

^=0.940

R _training ²

^=0.956

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

PSfrag replaements

CSST -

T

eq, ψ/δ ˙ _H

Target

Neuralnetworkpredition

(b)

R ² _test

^=0.960

R ² _training

^=0.975

Figure6.4: Regressionplots for CV

a _y,max

^and

T _eq, ψ/δ ˙ H

Asweanseefromgure6.3andgure6.4,themodelqualityofthetraining

data (

R _training ²

^{) is larger than the the model quality of the test data (}

R ² _test

^),

whih means, there is no overtting. The model quality of all omputed

re-sponse surfaes for all CVs is more than 0.9, whih means, the requirement

identied inthe last hapterregarding the model quality ismet. The

mislas-siation of eahSVM for eahCV islisted below:

Table 6.3: Mislassiation ofall trained SVMfor eah CV

CV Constraints of CVs

EG _H,nl

^1.1

a y,max

^1.1

( ˙ ψ/δ _H ) _stat,70

^1.1

( ˙ ψ/δ _H ) _stat,190

⁸

( ˙ ψ/δ H ) stat,max

^4.2

β _max

^12.8

F _z,min

^13.7

T _eq, ψ/δ ˙ H

15

(H/H ₀ ) _{ψ/δ ˙}

H

11.5

T _{a y /δ H}

¹²

∆ ˙ ψ _1s

^12.1

All the mislassiations are less than 15; hene the requirement regarding

themislassiation introdued inthe last hapter issatised.

Inthethirdstep,wehavetosetonstraintsontheCVs. Bysettinglowerand

upper bounds on eah CV, the solution spae will be formed. As mentioned

before, the largest volume of the solution box is unique. But there ould be

more than one box in the solution spae whih has the largest volume. For

instane, let uslook atthesolution spaeof twodesign variables,theatuator

transportdelay

T _d

^{,andthenaturalfrequenyfatorofthedynami}feedforward ontrol

w _f

^,gure6.5.

Figure 6.5:Solution spae of the atuator's transport delay and natural

fre-queny fatorof thedynamifeedforward ontrol

Based on the introdued optimization algorithm inthe last hapter, the

al-gorithmseeksasolutionboxwiththemaximumsizeanddoesnot onsiderthe

meaning ofdesign variables. Forinstane, thesolutionboxillustrated ingure

6.6 restrits the lower bound of the transport delay interval. It means, the

atuator transport delay mustnot begin fromzero, whih implies slow

perfor-maneoftheatuatorregardingitsinput. Butontheotherhand,theexibility

for the parameterization ofthe natural frequenyfator hasbeen improved by

extending its permissible interval with respet to this solution box. However,

suh a solution box is undesirable, as we require a high-performane atuator

preferablywithnodelayandofoursemoreexibilityfordesigninganatuator.

Figure 6.6:Solutionboxwiththe largestintervalfor thenatural frequeny

fa-tor ofthe dynami feedforward ontrol

Balaning design variables of the rear steering atuator and the

ontrol system logi

Therefore,thealgorithmhastobeforedtoavoidsuhundesirableomputation

eets. As a result, we have to balane the atuator and the ontrol system

logi designvariables before starting theoptimization proedure. Aordingly,

we xthe lower bound of theatuator transportdelay

T _d

^{to zeroinorder not}

to allow the optimization algorithm to nd a boxwhih restrits the atuator

performane, i.e. an interval for the transport delay is alulated whih does

not start fromzero. Basedon the samereason, we also xthelower boundof

thetimeonstant

T _w

^{oftheatuator transferfuntion. Thelowerboundofthe}

damping ratio

D

^{hasalso to be xedto 0.707 inorder to prevent a resonane}

magniation [46 ℄. In order to prevent limiting the maximal atuator power,

the upperboundsof parameters

v _noload

^and

F _max

^{havealsoto be xedto their}

dened uppervalues.

Consequently, all onstraints demanded for nding the largest box an be

summarized mathematially asfollows:

Ω⊆Ω max ds

µ(Ω)

subjetto







f _i ( x ) ≤ f _c,i , i = 1, ..., n ₁ g _j ( x ) = 1, j = 1, ..., n ₂ h _k ( x ) ≤ h _c,k , k = 1, ..., n ₃

for all

x ⊆ Ω

^(6.1)

where

f _i

^{isa predited funtion of the}

i

^{-th objetive driving dynamis}

per-formane measures by ANN,

g _j

^{is a lassiation of the}

j

^{-th objetive driving}

dynamis performane measures and

h _k

^{represents additional boundary}

on-straintsformulated on the

k

^{-thdesign variables.}

Now,bysettingalltheonstraintsandexeutingtheoptimization,thelargest

solution box is found for

f f d ≥ 95%

^{. Figure 6.7 shows the projetion of the}

omputed box for some design values based on the interative design spae

projetion and modiation. The volume of the omputed box is

7.5e ⁻¹⁰ %

of the entire design spae volume. At the rst glane, it seems to be very

small, but it still gives us exibility in the proedure of the atuator design

andthe adjustmentoftheontrolsystemparameters. Inaddition, suhasmall

volumeprovesthatndingthesolutionboxmanuallyisalmostimpossible. The

intervals of the design variables (normalized), obtained from thesolution box,

are depited ingure6.8 .

Figure6.7: Projetion ofthesolution boxfor some onsidereddesign variables

Figure 6.8:ARS Solution intervals(normalized)

The parameters of the dynami feedforward ontrol and the feedbak

on-troller an be adjusted now with respet to the determined intervals. The

harateristi urveoftheARSstatifeedforwardontrol analsobelaying in

theyellow areaofgure 6.9.

Figure 6.9:Solution spae for the harateristi urve of the ARS stati

feed-forward ontrol

Computing requirements on veriation test variables of ARS

atuator

Aswasexplainedintheprevioushapter, thesolutionintervalsoftheatuator

design variables should beonverted into requirements on theveriation test

variablesoftheatuator. Theserequirementsanbethenspeiedinthe

prod-utrequirementdoument (PRD) oftheatuator,whihhasto besatisedby

thesupplier. Asmentionedbefore,theserequirementsareformulatedregarding

theveriation tests. The veriation testshave to be arriedout on the

test-rig depited ingure 6.10 . This has to be done for a seleted maximum rear

steering angle from its solution intervals. The following requirements will be

alulated for

δ _r,max = 2.7 ^◦

^{,themaximumoftherear steering angleinterval.}

Figure6.10: Test-rig setupof ARS

Requirements of the ARS atuator with respet to performane test

It is denitely possible to ompute solution intervals for the veriation test

variablesfromthesolutionintervalsofthedesignvariables aswell. However, it

is enough to dene the worst-ase of the atuator response to the veriation

tests with respet to the alulated solution intervals. Before alulating the

worst-aseofthe atuator fore-veloityharateristi urve, ithastobe

men-tioned thattheupperboundofthisdesignvariable(

F _max

^,

v _noload

^)wasxedto

a onstant value during the optimization proess. This onstant value should

also begiven tosuppliers,astheymaynot onstrutanatuator withthe

per-formanemorethanthisonstantvalue. Jumpingbaktothealulationofthe

requirements on the veriation test variables, the worst-ase of the atuator

fore-veloity harateristi urve ours by the lower bounds of the solution

intervals of the

F _max

^,

v _noload

^and

v _{(F max )}

^{. Figure 6.11 depits the minimum}

ARS atuator fore-speed harateristi urve. The redarea is theprohibited

area. It means, a supplier must onstrut an atuator whose fore-veloity

harateristi urve isloated above this area.

Figure6.11: MinimumARS atuator fore-veloityharateristi urve

Requirements of the ARS atuator with respet to step response test

Generally,itisdesirabletohavelessovershoot,lessrisingtime,lessstabilization

time, andlesstime delay(quik responseto the stepinput). Thefatis, every

mehanial omponent, e.g. atuator, denitely has some overshoot, rising

time, stabilization time and time delay beause of it's physial onstrution

limits. Asa onsequene, the worst-ase of theatuator response to thestep

input with respet to the alulated solution intervals leads to the maximum

overshoot,maximumtimedelay,maximumstabilizationtime,maximumrising

time,andmaximumstationaryerror. Alltheseworst-asessubsequentlysatisfy

theupperand lowerboundsofthe design variables solutionintervals.

Insertingthe lowerbound ofthedamping ratiointerval

D

^{into equation5.15}

leads tothemaximumovershoot. Inthismanner,itguaranteesthesatisfation

D

Thestabilization time isdependent on the damping ratio andthetime

on-stant regardingequation5.16 . Aordingly,themaximumstabilization timeis

alulated fromthelowerbound ofthe determined interval ofthedamping

ra-tio,

D

^{,andthe upperboundofthetime onstant,}

T _w

^{,solutioninterval;hene,}

themaximumstabilizationtime guarantees therealizationoftheupperbound

of the time onstant solutioninterval.

With respet to equation 5.17 , the worst-ase of the rising time ours by

the upper bounds of the damping ratio and the time onstant interval.

Con-sequently, the maximum rising time realizes the upper bound of the solution

intervals of

D

^and

T _w

^.

Moreover, it is desirable to have a stationary error as small as possible. As

a result, the maximum possible permitted stationary error hasto be given to

suppliers. Thisarisesout oftheminimumbetween theratios oftheupperand

lowerbound of thesolutionintervalofthe stationarygain

K _p

^and1.

Finally, the maximum time delay is measured by the upper bound of the

determined intervals of

T _d

^{. Here, it should again be mentioned that thelower}

bound ofthe

T _d

^{wasxedduring the}optimization.

It should also be notied that alulating all the worst-ase requirements

has been arried out regarding worst-ase of the fore-veloity harateristi

urve. In other words, after inserting all these upper and lower bounds of the

determined intervalsinto thetransferfuntionoftheARSatuator andsetting

thestepasinput,theworst-aseofthefore-veloityshouldbebuiltasthe

rate-limiteraftertheoutputofthetransferfuntion,showningure4.17 . Then,all

the requirementsareformulated for theoutput asfollows:

Table 6.4: ARSatuator step responserequirements

Fore speed urve parameters Value

Max. Time delay

25ms

Max. Overshoot

0.0%

Max. Stabilization Time

405ms

Max. RisingTime

319ms

Max. Stationary Error

0.3%

Requirements of the ARS atuator with respet to sine test

Here, the test is dened with an inreasing frequeny from 0.1 to 5 Hz. The

amplitude is onsidered as

1 ^◦

^{. As mentioned in the previous hapter, the}

re-quirements shouldbeformulatedon thebodediagram. It isdesirable thatthe

deayofthemagnitudeandthephaseshifttakesplaeinhigherfrequeny,but

not inanyhighfrequeny. Inthis manner,the atuator performs wellin lower

frequenies and the output of the atuator has no phase shift and amplitude

deay in low frequenies. As a onsequene, therequirements of this test are

dened for the upper and lower bound of the bode diagram. The better the

phase response, the better the performane of the atuator in the frequeny

domain is. Therefore, the phase response of the atuator may not be worse

than the determined worst-ase. Itmeans, the frequeny responseof theARS

atuator shouldbe equal or better than this worst response. The deayof the

magnitude and the phaseshift of the bode diagram ofa seond-order transfer

funtion ours at the natural frequeny of the system [45 ℄. The natural

fre-queny isequal to

ω ₀ = _T ¹

w

. Inother words, the smallerthenatural frequeny

(the largerthe timeonstant)is, thesoonerthe dereaseof theamplitudeand

phase shift takes plae. Asa onsequene,the worst-aseof thebode-diagram

takesplae on the upperbound ofthe optimized timeonstant interval.

Con-erning the damping ratio

D

^{, the larger the damping ratio is, the earlier the}

magnitude response dereases. Subsequently, the worst-ase of the bode

dia-gramoursbytheupperboundoftheoptimizeddampingratiointerval;hene,

theworst-aseof thebode-diagram realizes the upperboundsofthe

D

^and

T _w

solution intervals. However, as mentioned before, thedeay ofmagnitude and

phase shift may not our at ordinary high frequeny. So, the lower bounds

of

D

^and

T _w

^{resultinthe upperbound ofthe} bode-diagram. Aordingly, the permitted bode-diagram isbetween the two redareas depited gure6.12 .

Figure6.12: Frequeny response requirement for theARS atuator

Im Dokument On the Design of Actuator and Control Systems in Early Development Stages (Seite 134-163)